According to the classical characterization theorems of mathematical statistics, the Gaussian distribution on the real line is characterized by the independence of two linear forms of independent random variables (the Skitovich–Darmois theorem), as well as by the symmetry of the conditional distribution of one linear form given another (the Heyde theorem). A number of papers have been devoted to generalizing of these theorems to various algebraic structures,
in particular, to locally compact Abelian groups (see e.g. [1]}, [2]}, [3]}, [4]}, , [5]}, , [6]}, [7]}, [8]}, and also [9]}).
In so doing, the coefficients of linear forms are topological automorphisms of the group. In this note, we consider a group analog of the generalized Pólya theorem, in which the Gaussian distribution on the real line is characterized by the property of equidistribution of a monomial and a linear form of independent identically distributed random variables. We study the generalized Pólya theorem on \(a\)-adic
solenoids. This is an important class of locally compact Abelian groups, since each connected locally compact Abelian group of dimension 1 is topologically isomorphic to either the additive group of real numbers, or the circle group, or an \(a\)-adic
solenoid.
Furthermore, the coefficients of a monomial and a linear form, as in the group analogs of the Skitovich–Darmois and Heyde theorems, are topological automorphisms of the \(a\)-adic
solenoid. As far as we know, such problem for groups has not been considered earlier. Note also that in the list of unsolved problems in [10]} the problem of constructing a theory of equidistributed linear forms on algebraic structures was formulated. A step in this direction is this note.
